Algebraic exponentiation for categories of Lie algebras
type: Item
instance of: scholarly article
title: Algebraic exponentiation for categories of Lie algebras
zbMATH Open document ID: 1275.18021
author: James Richard Andrew Gray
published in: Journal of Pure and Applied Algebra
publication date: 2012-10-16T00:00:00Z
reviewed by: Laurent Poinsot
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 18C20
zbMATH DE Number: 6094157
zbMATH Keywords: Lie algebra
zbMATH Keywords: pullback functor
zbMATH Keywords: category of points
zbMATH Keywords: locally Cartesian closed category
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.02.034
OpenAlex ID: W2075013969
Wikidata QID: Q115345495
cites work: Q3978968
cites work: Q3130029
cites work: On exponentiability of étale algebraic homomorphisms
cites work: Algebraic exponentiation in general categories
cites work: Cartesianness: topological spaces, uniform spaces, and affine schemes
DOI: 10.1016/J.JPAA.2012.02.034
review text: A category is locally Cartesian closed if, and only if, every pullback functor between slice categories has a right adjoint. Any such pullback functor lifts to the corresponding category of points (the category of split epimorphisms with fixed codomain), and the existence of a right adjoint for the pullback functor implies that its lifting also admits a right adjoint. The category of Lie algebras over a commutative ring \(D\) with a unit is not locally Cartesian closed, and the author considers thus the problem of the existence of a right adjoint to the pullback functor between categories of points of Lie algebras over \(D\). The category of points over such a Lie algebra may be identified with a category of \(D\)-modules which are also modules over the Lie algebra (with compatibility).NEWLINENEWLINE The author proves that the obvious forgetful functor from this category of modules and that of Lie algebras over \(D\) admits a right adjoint which is even explicitly constructed. From that result he deduces the existence of a right adjoint to every pullback functors between categories of points of Lie algebras over \(D\).
description: scientific article; zbMATH DE number 6094157